| Title:
|
Remarks on fixed points of rotative Lipschitzian mappings (English) |
| Author:
|
Górnicki, Jarosław |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
40 |
| Issue:
|
3 |
| Year:
|
1999 |
| Pages:
|
495-510 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $C$ be a nonempty closed convex subset of a Banach space $E$ and \linebreak $T:C\rightarrow C$ a $k$-Lipschitzian rotative mapping, i.e\. such that $\|Tx-Ty\|\leq k\cdot \|x-y\|$ and $\|T^n x-x\|\leq a\cdot \|x-Tx\|$ for some real $k$, $a$ and an integer $n>a$. The paper concerns the existence of a fixed point of $T$ in $p$-uniformly convex Banach spaces, depending on $k$, $a$ and $n=2,3$. (English) |
| Keyword:
|
rotative mappings |
| Keyword:
|
fixed points |
| MSC:
|
47H09 |
| MSC:
|
47H10 |
| idZBL:
|
Zbl 1065.47504 |
| idMR:
|
MR1732485 |
| . |
| Date available:
|
2009-01-08T18:54:37Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119106 |
| . |
| Reference:
|
[1] Barros-Neto J.: An Introduction to the Theory of Distributions.M. Dekker, New York, 1973. Zbl 0512.46040, MR 0461128 |
| Reference:
|
[2] Dunford N.: Linear Operators.vol. I, Interscience, New York, 1958. Zbl 0635.47003 |
| Reference:
|
[3] Duren W.L.: Theory of $H^p$ Spaces.Academic Press, New York, 1970. MR 0268655 |
| Reference:
|
[4] Goebel K.: Convexity of balls and fixed point theorems for mappings with nonexpansive square.Compositio Math. 22 (1970), 269-274. Zbl 0202.12802, MR 0273477 |
| Reference:
|
[5] Goebel K., Kirk W.A.: Topics in Metric Fixed Point Theory.Cambridge Studies in Advanced Mathematics 28, Cambridge University Press, London, 1990. MR 1074005 |
| Reference:
|
[6] Goebel K., Koter M.: A remark on nonexpansive mappings.Canad. Math. Bull. 24 (1981), 113-115. Zbl 0461.47027, MR 0611220 |
| Reference:
|
[7] Goebel K., Koter M.: Fixed points of rotative Lipschitzian mappings.Rend. Sem. Mat. Fis. di Milano 51 (1981), 145-156. Zbl 0535.47031, MR 0708040 |
| Reference:
|
[8] Goebel K., Złotkiewicz E.: Some fixed point theorems in Banach spaces.Colloquium Math. 23 (1971), 103-106. MR 0303367 |
| Reference:
|
[9] Górnicki J.: Fixed points of involutions.Math. Japonica 43 (1996), 151-155. MR 1373993 |
| Reference:
|
[10] Górnicki J., Rhoades B.E.: A general fixed point theorem for involutions.Indian J. Pure Appl. Math. 27{(1)} (1996), 13-23. MR 1374884 |
| Reference:
|
[11] Kirk W.A.: A fixed point theorem for mappings with a nonexpansive iterate.Proc. Amer. Math. Soc. 29 (1971), 294-298. Zbl 0213.41303, MR 0284887 |
| Reference:
|
[12] Komorowski T.: Selected topics on Lipschitzian mappings (in Polish).Thesis, Univ. M. Curie-Sklodowskiej, Lublin, 1987. |
| Reference:
|
[13] Koter M.: Fixed points of Lipschitzian $2$-rotative mappings.Bolletino U.M.I., Ser. VI, 5 (1986), 321-339. Zbl 0634.47053, MR 0897203 |
| Reference:
|
[14] Lim T.C., Xu H.K., Xu Z.B.: An $L^p$ inequality and its applications to fixed point theory and approximation theory.in: Progress in Approximation Theory (P. Nevai and A. Pinkus, eds.), Academic Press, New York, 1991, pp.609-624. Zbl 0801.46012, MR 1114800 |
| Reference:
|
[15] Lindenstrauss J., Tzafriri L.: Classical Banach Spaces, II - Function Spaces.Springer-Verlag, Berlin, 1979. Zbl 0403.46022, MR 0540367 |
| Reference:
|
[16] Linhart J.: Fixpunkte von Involutionen $n$-ter Ordnung.Österich. Acad. Wiss. Math.-Natur. kl. S-B 180 (1973), 89-93. MR 0303369 |
| Reference:
|
[17] Prus B., Smarzewski R.: Strongly unique best approximations and centers in uniformly convex spaces.J. Math. Anal. Appl. 121 (1987), 10-21. Zbl 0617.41046, MR 0869515 |
| Reference:
|
[18] Smarzewski R.: Strongly unique minimization of functionals in Banach spaces with applications to theory of approximation and fixed points.J. Math. Anal. Appl. 115 (1986), 155-172. Zbl 0593.49004, MR 0835591 |
| Reference:
|
[19] Smarzewski R.: Strongly unique best approximation in Banach spaces, II.J. Approx. Theory 51 (1987), 202-217. Zbl 0657.41022, MR 0913618 |
| Reference:
|
[20] Smarzewski R.: On the inequality of Bynum and Drew.J. Math. Anal. Appl. 150 (1990), 146-150. MR 1059576 |
| Reference:
|
[21] Xu H.K.: Inequalities in Banach spaces with applications.Nonlinear Anal. 16 (1991), 1127-1138. Zbl 0757.46033, MR 1111623 |
| Reference:
|
[22] Zălinescu C.: On uniformly convex function.J. Math. Anal. Appl. 95 (1983), 344-374. MR 0716088 |
| . |
